Introduction Linear example Nonlinear example Conclusion Ensemble Control as a Tool for Robot Motion Planning: Uncertainty, Optimality, and Complexity Timothy Bretl (w/ Aaron Becker) University of Illinois at Urbana-Champaign IEEE ICRA Workshop on Uncertainty in Automation May 9, 2011 tbretl@illinois.edu Ensemble control for robot motion planning 1/ 32
Introduction Linear example Nonlinear example Conclusion Problem Message Outline Introduction 1 Problem Message Linear example 2 Nonlinear example 3 Conclusion 4 tbretl@illinois.edu Ensemble control for robot motion planning 2/ 32
Introduction Linear example Nonlinear example Conclusion Problem Message Motion planning Find u : [0 , t f ] → U ⊂ R m x : [0 , t f ] → X ⊂ R n satisfying x ( t ) = f ( t , x ( t ) , u ( t )) ˙ x (0) = x start x ( t f ) = x goal for free final time t f tbretl@illinois.edu Ensemble control for robot motion planning 3/ 32
Introduction Linear example Nonlinear example Conclusion Problem Message Motion planning under bounded uncertainty Find u : [0 , t f ] → U ⊂ R m x : [0 , t f ] → X ⊂ R n satisfying x ( t ) = f ( t , x ( t ) , u ( t ) , ǫ ) ˙ x (0) = x start x ( t f ) = x goal for free final time t f , despite bounded uncertainty ǫ ∈ [1 − δ, 1 + δ ] tbretl@illinois.edu Ensemble control for robot motion planning 4/ 32
Introduction Linear example Nonlinear example Conclusion Problem Message Motion planning as ensemble control Find u : [0 , t f ] → U ⊂ R m x : [0 , t f ] × [1 − δ, 1 + δ ] → X ⊂ R n satisfying x ( t , ǫ ) = f ( t , x ( t , ǫ ) , u ( t ) , ǫ ) ˙ x (0 , ǫ ) = x start x ( t f , ǫ ) = x goal for free final time t f and for all ǫ ∈ [1 − δ, 1 + δ ] tbretl@illinois.edu Ensemble control for robot motion planning 5/ 32
Introduction Linear example Nonlinear example Conclusion Problem Message Take-away message Ensemble control theory is a useful way to deal with bounded uncertainty in dynamical systems. To steer one system with an uncertain parameter, we pretend to steer a continuous ensemble of systems, each with a particular value of that parameter. In the examples we will consider, this approach costs us nothing in terms of computational complexity. tbretl@illinois.edu Ensemble control for robot motion planning 6/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Outline Introduction 1 Linear example 2 Model Analysis Results Nonlinear example 3 Conclusion 4 tbretl@illinois.edu Ensemble control for robot motion planning 7/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results A driven harmonic oscillator • This system is linear and has the form � 0 � � 0 � m 1 x = ˙ x + u y − ǫ 0 ǫ k = Ax + Bu , where ( x 1 , x 2 ) = ( y , ˙ y ), u = d , and ǫ = k / m . d • For known ǫ , this system is controllable because � ǫ � 0 � � AB B = 0 ǫ is full rank. tbretl@illinois.edu Ensemble control for robot motion planning 8/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Ensemble controllability • For unknown ǫ , consider the lifted system � 0 � � 0 � 1 x ( t , ǫ ) = ˙ x ( t , ǫ ) + u ( t ) − ǫ 0 ǫ = A ( ǫ ) x ( t , ǫ ) + B ( ǫ ) u ( t ) . • For any integer k ≥ 0, we have � ǫ k +1 � 0 � A 2 k +1 B A 2 k B � = . ǫ k +1 0 • So, we can approximate any desired movement direction by a polynomial in ǫ , with error vanishing in k : k − 1 � � 1 � � 0 �� � ǫ i f ( ǫ ) ≈ a i + b i 0 1 i =0 tbretl@illinois.edu Ensemble control for robot motion planning 9/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Control by piecewise-constant inputs (1/2) • A discrete-time model is �� T � x d ( i + 1 , ǫ ) = e A ( ǫ ) T x d ( i , ǫ ) + e As Bds u d ( i ) 0 = A d ( ǫ ) x d ( i , ǫ ) + B d ( ǫ ) u d ( i ) • If x d (0 , ǫ ) = 0 then 2 k � A i x d (2 k , ǫ ) = d ( ǫ ) B d ( ǫ ) u d (2 k − i ) i =0 • This can be approximated by the series expansion k − 1 � ∂ i x d (2 k , ǫ ) 1 � � � ( ǫ − 1) i � x d (2 k , ǫ ) ≈ � i ! ∂ǫ i � ǫ =1 i =0 tbretl@illinois.edu Ensemble control for robot motion planning 10/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Control by piecewise-constant inputs (2/2) • The result has the form k � r i � ( ǫ − 1) i − 1 + O � | ǫ − 1 | k � � x d (2 k , ǫ ) = s i i =1 where r , s ∈ R k are linear in u d ∈ R 2 k • To achieve ( x 1 , x 2 ) with error of order k in | ǫ − 1 | : x 1 x 2 0 0 r = , s = . . . . . . 0 0 • The solution (2 k linear equations in 2 k variables) has the form u d = K 1 x 1 + K 2 x 2 for matrices K 1 and K 2 that can be precomputed tbretl@illinois.edu Ensemble control for robot motion planning 11/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Example results in simulation tbretl@illinois.edu Ensemble control for robot motion planning 12/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Outline Introduction 1 Linear example 2 Nonlinear example 3 Model Analysis Results Conclusion 4 tbretl@illinois.edu Ensemble control for robot motion planning 13/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results A differential drive robot with uncertain wheel radius For a fixed wheel separation, inputs scale with wheel radius ǫ r = wheel radius b = wheel separation � r ( ω R + ω L ) � v = ǫ = ǫ u 1 2 � r ( ω R − ω L ) � w = ǫ = ǫ u 2 b ǫ = 0 . 8 ǫ = 0 . 83 ǫ = 1 . 0 ǫ = 1 . 2 tbretl@illinois.edu Ensemble control for robot motion planning 14/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Scratch-drive microrobots with uncertain forward speed Figure: Donald et al. (2006) • For a fixed turning radius, inputs scale Figure: Donald et al. (2008) with forward speed tbretl@illinois.edu Ensemble control for robot motion planning 15/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Analysis of one robot • This system is nonlinear and has the form cos x 3 0 u 1 + ǫ u 2 x = ǫ ˙ sin x 3 0 0 1 = ǫ g 1 ( x ) u 1 + ǫ g 2 ( x ) u 2 • For known ǫ > 0, this system is controllable because ǫ 2 sin x 3 0 ǫ cos x 3 − ǫ 2 cos x 3 � � [ ǫ g 1 , ǫ g 2 ] ǫ g 2 ǫ g 1 = 0 ǫ sin x 3 0 ǫ 0 is full rank everywhere tbretl@illinois.edu Ensemble control for robot motion planning 16/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Analysis of an ensemble (1/3) • For unknown ǫ , consider the lifted system cos x 3 ( t , ǫ ) 0 u 1 ( t ) + ǫ u 2 ( t ) x ( t , ǫ ) = ǫ ˙ sin x 3 ( t , ǫ ) 0 0 1 = ǫ g 1 ( x ( t , ǫ )) u 1 ( t ) + ǫ g 2 ( x ( t , ǫ )) u 2 ( t ) • Heading is not controllable, since for all ǫ ˙ x 3 ( t , ǫ ) = x 3 (0 , ǫ ) + ǫθ ( t ) where θ ( t ) = u 2 ( t ) • Eliminate heading to get a controllable subsystem: x 1 ( t , ǫ ) ˙ cos ( x 3 (0 , ǫ ) + ǫθ ( t )) 0 = ǫ u 1 ( t ) + u 2 ( t ) x 2 ( t , ǫ ) ˙ sin ( x 3 (0 , ǫ ) + ǫθ ( t )) 0 ˙ θ ( t ) 0 1 tbretl@illinois.edu Ensemble control for robot motion planning 17/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Analysis of an ensemble (2/3) • Take Lie brackets to find new control vector fields: � ∂ g 2 ∂ q g 1 − ∂ g 1 � [ ǫ g 1 , g 2 ] = ǫ ∂ q g 2 − sin ( x 3 (0 , ǫ ) + ǫθ ( t )) = − ǫ 2 cos ( x 3 (0 , ǫ ) + ǫθ ( t )) 0 = − ǫ 2 g 3 [[ ǫ g 1 , g 2 ] , g 2 ] = − ǫ 3 g 1 [[[ ǫ g 1 , g 2 ] , g 2 ] , g 2 ] = − ǫ 4 g 3 . . . tbretl@illinois.edu Ensemble control for robot motion planning 18/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Analysis of an ensemble (3/3) • We can approximate any desired movement direction by a polynomial in ǫ , with error vanishing in k : k � a i ǫ 2 i +1 g 1 + b i ǫ 2 i +2 g 3 � � f ( ǫ ) ≈ cg 2 + i =0 tbretl@illinois.edu Ensemble control for robot motion planning 19/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Control with piecewise-constant inputs ∆ t u 1 u 2 tbretl@illinois.edu Ensemble control for robot motion planning 20/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Control with piecewise-constant inputs ∆ t u 1 u 2 0 1 /λ λ ( j − 1) φ ( j − 1) φ tbretl@illinois.edu Ensemble control for robot motion planning 20/ 32
Introduction Linear example Nonlinear example Conclusion Model Analysis Results Control with piecewise-constant inputs ∆ t u 1 u 2 0 1 /λ λ ( j − 1) φ a ′ � � � � a ′ � a ′ sign 0 j � � j j � tbretl@illinois.edu Ensemble control for robot motion planning 20/ 32
Recommend
More recommend
